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On Homotopy Spheres Is Considered transactions of the american mathematical society Volume 275, Number 2, February 1983 DIFFERENTIABLEGROUP ACTIONS ON HOMOTOPYSPHERES: III. INVARIANTSUBSPHERES AND SMOOTH SUSPENSIONS BY REINHARD SCHULTZ1 Abstract. A linear action of an abelian group on a sphere generally contains a large family of invariant linear subspheres. In this paper the problem of finding invariant subspheres for more general smooth actions on homotopy spheres is considered. Classification schemes for actions with invariant subspheres are obtained; these are formally parallel to the classifications discussed in the preceding paper of this series. The realizability of a given smooth action as an invariant codimension two sub- sphere is shown to depend only on the ambient differential structure and an isotopy invariant. Applications of these results to specific cases are given; for example, it is shown that every exotic 10-sphere admits a smooth circle action. In our previous papers in this series [45,44], we have considered the theory of semifree actions on homotopy spheres as formulated by W. Browder and T. Pétrie [10] and M. Rothenberg and J. Sondow [34]. Specifically, in the first paper a method was presented for describing (at least formally) those exotic spheres admitting such semifree actions—a problem first posed explicitly by Browder in [3, Problem l,p. 7] —and the second paper extended the whole theory to handle certain actions that are not semifree. This paper will treat another problem posed in Browder's paper [3, Problem 3] regarding invariant subspheres of homotopy spheres with group actions. One motivation for considering this question is that linear actions on spheres generally admit a great assortment of invariant linear subspheres (e.g., if the group is abelian and the dimension is much larger than the group's order), and from this viewpoint the existence of invariant subspheres reflects the extent to which an arbitrary smooth action resembles some natural linear model. In particular, this idea is central to the work of Browder and Livesay on free involutions [8] (compare also [3]). The existence of such subspheres is directly related to the realizability of actions as equivariant smooth suspensions, providing basic necessary conditions for such a realization. We shall also consider a dual problem in this paper; namely, the description of those group actions that can be smoothly equivariantly suspended. Questions of this sort first arose in the study of free involutions [16], and their close Received by the editors August 5, 1981 and, in revised form, February 2, 1982. 1980 MathematicsSubject Classification.Primary 57S15, 57S25; Secondary 57R40, 57R52, 57R55, 57R65. 'Partially supported by NSF Grants MCS76-08794and MCS78-02913A. ©1983 American Mathematical Society 0OO2-9947/82/0O0O-0328/$06.50 729 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 730 REINHARD SCHULTZ relationship with the invariant subsphere problem was clarified in later work of Browder and Livesay [8], Lopez de Medrano [29], and others. Of course, there is no topological obstruction to suspending an action, and given that exotic spheres roughly represent the simplest differences between the DIFF and TOP categories, it is fairly predictable that the obstructions should involve groups of homotopy spheres. Actually, these obstructions already arose in [45 and 49]; for circle actions, the differential structure on the homotopy sphere is the obstruction to smooth suspendabihty, while for cychc group actions one must supplement a differential structure condition with an assumption that the action is isotopically trivial. We shall find it convenient to divide invariant homotopy subspheres into two types. The Type I (homotopy) subspheres are those invariant içS such that the action is free on 2 — K, while the Type II subspheres are more or less the others. An important advantage of Type I subspheres is that the methods of [50] adapt easily to discuss them. Aside from its intrinsic interest, the invariant subsphere question—especially for Type I subspheres—and the equivariant suspension construction figure importantly in answering some basic questions about the smooth S ' and Zp actions that can exist on a given exotic sphere. To illustrate the apphcability of these notions, we shall construct a smooth S1 action on the generator of 01O = Z6, which previously had not been shown to admit such an action (compare [41]). The results on invariant subspheres in this paper also suffice to complete the proofs of some assertions in [44] about realizing certain classes in irxiDiîf(S")) by continuous representations of the circle group (see §4). More comprehensive apphcations will appear in papers IV-VI of this series, including (i) a purely homotopy-theoretic characterization of those exotic spheres admitting Zp ( p an odd prime) actions with a fixed point set of given codimension and (ii) an inductive method for constructing smooth circle actions on exotic spheres with Pontrjagin-Thom invariants ß2,... ,ßp_x G ir^P) (see [48] for a partial summary). There have been several previous studies of invariant subspheres, particularly in the free case (e.g., see [3,8,9,12,13,17,18,19,25,29,55]). For the most part, such studies have dealt with a class of invariant subspheres called characteristic (see §2 for a description in our context; also compare [3], for example). Characteristic sub- spheres of semifree and more general actions have also been studied in [3,9, and 56]. I am extremely grateful to Wu-chung Hsiang, Ted Pétrie, and Valdis Vijums for discussing with me results of theirs which were unpublished, at least in 1976-1977 when the discussions took place. I would also like to thank Princeton University, the Institute for Advanced Study, the University of Chicago, and the University of Michigan for their hospitality while the manuscript for this paper was written. 1. Type I subspheres and their knot invariants. We are mainly interested in the theory and applications of invariant subspheres for semifree actions. However, it is just as easy to formulate everything for ultrasemifree actions as defined in [50, p. 267]. If G is abelian, an effective action is ultrasemifree if the set of isotropy subgroups Gx ¥^ {1} has a unique minimal element (called the subprincipal isotropy License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use DIFFERENTIABLE GROUP ACTIONS ON HOMOTOPY SPHERES 731 subgroup). In particular, Up is prime then every action of Zpr is ultrasemifree. If G acts semifreely, then G itself is the subprincipal isotropy subgroup; in general, if H is the subprincipal isotropy subgroup, then G acts freely on the complement of the fixed point set of //. As stated in the introduction, we shall be interested in triples 2 D K Z) F of smooth homotopy spheres, where (a) G = Zpr ip prime), Qr (generalized quaternion group), S\ or S3 acts smoothly and ultrasemifreely on 2 with subprincipal isotropy bound (or group) H. (b) K is a (/-invariant homotopy subsphere. (c) F= KH = 2". This will be the precise definition of an ultrasemifree action on 2 with an invariant Type I i homotopy) subsphere. In order to organize our thoughts more efficiently, we assume x E F is a fixed point and denote the equivariant tangent spaces at x in F, K, and 2 by a, a + W, and a + V respectively; the G-modules W and V are free G-modules, and we make the allowable assumption W E V. Denote the quotient module V/W by Q. Since the triples in question all admit invariant Riemannian metrics with K a totally geodesic submanifold and the decomposition a + W + Ü may be assumed orthogonal, we shall assume such metrics have been chosen when necessary. Of course, a triple 2 D K D F defines two separate ultrasemifree G-actions, one on 2 and one on K. Therefore by the results of [45] one has two knot invariants associated to F — 2W = KH. It is natural to look for a relationship between these two knot invariants. An obvious model for such a relationship is given by the normal bundles associated to 2 D K D F; namely, the normal bundle viF, 2) is the sum of viF, K) plus the restriction of viK, 2) to F (compare [21]). In analogy with the situation for normal bundles, we need a third knot invariant, that of K in 2. Let f denote the normal bundle of K; then the arguments of [45,50] show that the composite (1.1) S^fi) — S(f ) Ç 2 — K {S = unit sphere or unit sphere bundle) is a G-homotopy equivalence if dimRi2 > 3 and a G-homotopy retract if dimRß = 2. Thus, as in [45,50], we may again define an equivariant fiber retraction S(f) -» S(fl), and we shall denote this by u0iK E 2). This invariant takes values in the set (1.2) F/0GQMeiK) (see [41,44] for notation). Remark. If F, K, and 2 are merely Z^-homology spheres, in analogy with [45] one may define a knot invariant in F(p)/0G fi freerather than F/0C a free. We can now state and prove a Sum Formula for knot invariants which parallels the well-known identity for normal bundles mentioned above. Theorem 1.3. Suppose that G as above acts ultrasemifreely on the triple 2 D K D F with 2H = F = KH. Let i: F -» K denote inclusion. Then the knot invariants of F in 2 and K are related by the formula to(F E 2) = co(F E K) © i*u>0iK E 2), where direct sum is defined as in [50, §2], License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 732 REINHARD SCHULTZ Remark.
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